2.4: Nonlinear Equations in $\mathbb{R}^n$

Here we consider the nonlinear differential equation
$$\begin{equation} \label{nonlinear2} F(x,z,p)=0, \end{equation}$$
where
$$x=(x_1,\ldots,x_n),\ z=u(x):\ \Omega\subset\mathbb{R}^n\mapsto\mathbb{R}^1,\ p=\nabla u.$$
The following system of $2n+1$ ordinary differential equations is called characteristic system.
$$\begin{eqnarray*} x'(t)&=&\nabla_pF\\ z'(t)&=&p\cdot\nabla_pF\\ p'(t)&=&-\nabla_xF-F_zp. \end{eqnarray*}$$
Let
$$x_0(s)=(x_{01}(s),\ldots,x_{0n}(s)),\ s=(s_1,\ldots,s_{n-1}),$$
be a given regular (n-1)-dimensional $C^2$-hypersurface in $\mathbb{R}^n$, i. e., we assume
$$\mbox{rank}\frac{\partial x_0(s)}{\partial s}=n-1.$$
Here $s\in D$ is a parameter from an $(n-1)$-dimensional parameter domain $D$.

For example, $x=x_0(s)$ defines in the three dimensional case a regular surface in $\mathbb{R}^3$.

Assume
$$z_0(s):\ D\mapsto\mathbb{R}^1,\ p_0(s)=(p_{01}(s),\ldots,p_{0n}(s))$$
are given sufficiently regular functions.

The $(2n+1)$-vector
$$(x_0(s),z_0(s),p_0(s))$$
is called initial strip manifold and the condition
$$\frac{\partial z_0}{\partial s_l}=\sum_{i=1}^{n-1}p_{0i}(s)\frac{\partial x_{0i}}{\partial s_l},$$
$l=1,\ldots,n-1$, strip condition.

The initial strip manifold is said to be non-characteristic if
$$\det\left(\begin{array}{llcl}F_{p_1}&F_{p_2}&\cdots & F_{p_n}\\ \frac{\partial x_{01}}{\partial s_1}&\frac{\partial x_{02}}{\partial s_1}&\cdots & \frac{\partial x_{0n}}{\partial s_1}\\ ... & ... & ... & ...\\ \frac{\partial x_{01}}{\partial s_{n-1}}&\frac{\partial x_{02}}{\partial s_{n-1}}&\cdots & \frac{\partial x_{0n}}{\partial s_{n-1}}\end{array}\right)\not=0,$$
where the argument of $F_{p_j}$ is the initial strip manifold.

Initial value problem of Cauchy. Seek a solution $z=u(x)$ of the differential equation ($\ref{nonlinear2}$) such that the initial manifold is a subset of $\{(x,u(x),\nabla u(x)):\ x\in \Omega\}$.

As in the two dimensional case we have under additional regularity assumptions

Theorem 2.3. Suppose the initial strip manifold is not characteristic and satisfies differential equation ($\ref{nonlinear2}$), that is,
$F(x_0(s),z_0(s),p_0(s))=0$. Then there is a neighborhood of the initial manifold $(x_0(s),z_0(s))$ such that there exists a unique solution of the Cauchy initial value problem.

Sketch of proof. Let
$$x=x(s,t),\ z=z(s,t),\ p=p(s,t)$$
be the solution of the characteristic system and let
$$s=s(x),\ t=t(x)$$
be the inverse of $x=x(s,t)$ which exists in a neighborhood of $t=0$. Then, it turns out that
$$z=u(x):= z(s_1(x_1,\ldots,x_n),\ldots,s_{n-1}(x_1,\ldots,x_n),t(x_1,\ldots,x_n))$$
is the solution of the problem.

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.